First slide

Binomial theorem for positive integral Index

Question

The coefficient of x²⁸ in the expansion of ${(1 + x^{3} - x^{6})}^{30}$ is

Very Easy

A

1
B

0
C

$^{30} C_{6}$
D

$^{30} C_{3}$

Solution

$\begin{array}{l} {(1 + x^{3} - x^{6})}^{30} \\ = {\{1 + x^{3} (1 - x^{3})\}}^{30} \\ =^{30} C_{0} +^{30} C_{1} x^{3} (1 - x^{3}) +^{30} C_{2} x^{6} {(1 - x^{3})}^{2} + \dots \end{array}$
Obviously, each term will contain $x^{3 m}, m \in N$ . But 28 is not divisible by 3. Therefore, there will be no term containing x²⁸.

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